
The reader may think that such a description of the world would be quite inadequate. What does it mean to assign to
an event the particular co-ordinates x1, x2, x3, x4, if in themselves these co-ordinates have no significance? More
careful consideration shows, however, that this anxiety is unfounded. Let us consider, for instance, a material point
with any kind of motion. If this point had only a momentary existence without duration, then it would to described in
space-time by a single system of values x1, x2, x3, x4. Thus its permanent existence must be characterised by an
infinitely large number of such systems of values, the co-ordinate values of which are so close together as to give
continuity; corresponding to the material point, we thus have a (uni-dimensional) line in the four-dimensional
continuum. In the same way, any such lines in our continuum correspond to many points in motion. The only
statements having regard to these points which can claim a physical existence are in reality the statements about their
encounters. In our mathematical treatment, such an encounter is expressed in the fact that the two lines which
represent the motions of the points in question have a particular system of co-ordinate values, x1, x2, x3, x4, in
common. After mature consideration the reader will doubtless admit that in reality such encounters constitute the only
actual evidence of a time-space nature with which we meet in physical statements.
When we were describing the motion of a material point relative to a body of reference, we stated nothing more than
the encounters of this point with particular points of the reference-body. We can also determine the corresponding
values of the time by the observation of encounters of the body with clocks, in conjunction with the observation of the
encounter of the hands of clocks with particular points on the dials. It is just the same in the case of space-
measurements by means of measuring-rods, as a little consideration will show.
The following statements hold generally: Every physical description resolves itself into a number of statements, each
of which refers to the space-time coincidence of two events A and B. In terms of Gaussian co-ordinates, every such
statement is expressed by the agreement of their four co-ordinates x1, x2, x3, x4. Thus in reality, the description of the
time-space continuum by means of Gauss co-ordinates completely replaces the description with the aid of a body of
reference, without suffering from the defects of the latter mode of description; it is not tied down to the Euclidean
character of the continuum which has to be represented.
28. EXACT FORMULATION OF THE GENERAL PRINCIPLE OF RELATIVITY
We are now in a position to replace the provisional formulation of the general principle of relativity given in Section
18 by an exact formulation. The form there used, "All bodies of reference K, K', etc., are equivalent for the description
of natural phenomena (formulation of the general laws of nature), whatever may be their state of motion," cannot be
maintained, because the use of rigid reference-bodies, in the sense of the method followed in the special theory of
relativity, is in general not possible in space-time description. The Gauss co-ordinate system has to take the place of
the body of reference. The following statement corresponds to the fundamental idea of the general principle of
relativity: "All Gaussian co-ordinate systems are essentially equivalent for the formulation of the general laws of
nature."
We can state this general principle of relativity in still another form, which renders it yet more clearly intelligible than
it is when in the form of the natural extension of the special principle of relativity. According to the special theory of
relativity, the equations which express the general laws of nature pass over into equations of the same form when, by
making use of the Lorentz transformation, we replace the space-time variables x, y, z, t, of a (Galileian) reference-
body K by the space-time variables x', y', z', t', of a new reference-body K'. According to the general theory of
relativity, on the other hand, by application of arbitrary substitutions of the Gauss variables x1, x2, x3, x4, the equations
must pass over into equations of the same form; for every transformation (not only the Lorentz transformation)
corresponds to the transition of one Gauss co-ordinate system into another.
If we desire to adhere to our "old-time" three-dimensional view of things, then we can characterise the development
which is being undergone by the fundamental idea of the general theory of relativity as follows : The special theory of
relativity has reference to Galileian domains, i.e. to those in which no gravitational field exists. In this connection a
Galileian reference-body serves as body of reference, i.e. a rigid body the state of motion of which is so chosen that
the Galileian law of the uniform rectilinear motion of "isolated" material points holds relatively to it.
Certain considerations suggest that we should refer the same Galileian domains to non-Galileian reference-bodies
also. A gravitational field of a special kind is then present with respect to these bodies (cf. Sections 20 and 23).